Conic Sections

Description

Conic Sections

If a plane intersects a cone the geometrical figures thus formed are called conic sections. A plane perpendicular to the base and passing through the vertex of a right circular cone forms an isosceles triangle. If the plane is parallel to the base the intersection of the plane and conical surfaces will be the circumference of a circle.

Ellipse. An ellipse is a curve formed by the intersection of a plane and a cone, Fig. 86a, or cylinder, Fig. 866, the plane being oblique to the axis but not cutting the base. An ellipse may be defined as a curve generated by a point moving in a plane in such a manner that the sum of the distances from the point to two fixed points shall always be constant.

The two fixed points are called foci, Fig. 87, and shall lie on the longest line that can be drawn in the ellipse which is called the major axis; the shortest line is called the minor axis, and is perpendicular to the major axis at its middle point, called the center.

Fig. 87. Ellipse.

Fig. 88. Parabola.

An ellipse may be constructed if the major and minor axes are given or if the foci and one axis are known.

Parabola. The parabola is a curve formed by the intersection of a cone and a plane parallel to an element of the cone, Fig. 86c. This curve is not a closed curve for the branches approach parallelism.

A parabola may be defined as a curve every point of which is equally distant from a line and a point.

The point is called the focus, Fig. 88, and the given line, the directrix. The line perpendicular to the directrix and passing through the focus is the axis. The intersection of the axis and the curve is the vertex.

Hyperbola. This curve is formed by the intersection of a plane and a cone, the plane being parallel to the axis of the cone, Fig. 86d.

Fig. 89. Hyperbola.

Like the parabola, the curve is not closed, the branches constantly diverging.

An hyperbola is defined as a plane curve such that the difference between the distances from any point in the curve to two fixed points is equal to a given distance.

The two fixed points are the foci and the line passing through them is the transverse axis, Fig. 89.

Rectangular Hyperbola. The form of hyperbola most used in Mechanical Engineering is called the rectangular hyperbola because it is drawn with reference to rectangular coordinates. This curve is constructed as follows: In Fig. 90, 0 X and 0 Y are the two coordinate axes drawn at right angles to each other. These lines are also called asymptotes. Assume A to be a known point on the curve. Draw A C parallel to 0 X and A D perpendicular to OX. Mark off any convenient points on A C such as E, F, G, and H, and through these points draw EE', FF', GG', and HH', perpendicular to 0 X. Connect E, F, G, H, and C with 0. Through the points of intersection of the oblique lines and the vertical line A D draw the horizontal lines LL', MM', NN', PP', and QQ'. The first point on the curve is the assumed point A, the second point is R, the intersection of LL' and EE'. The third is the intersection S of MM' and FF'. The other points are found in the same way.

In this curve the products of the coordinates of all points are equal. Thus LR X RE' = MS X SF' = NT X TG'.